How to calculate the optimal placements for settlements in Catan without an ML algorithm?
Catan is actually a much more complicated game than the simple rules would suggest, and an exact solution is probably beyond the scope of current AI techniques.

Monte Carlo Tree Search or Expectiminimax techniques seem like they could help, but are intended for games of perfect information. Catan is not a game of perfect information (the development cards are hidden), and also has a phase that occurs without a regular turn sequence (trading). 

To solve Catan properly, I think you're going to need both algorithms for solving POMDPs (like CFR+), and algorithms for negotiation (like Kraus' Diplomat). I'm not certain that these have been combined before in formal analysis, so this might actually be a good PhD thesis for someone.

That said, you can probably get a good player using self-play techniques, because Catan has randomization, and a relatively small set of moves, like Backgammon. These may or may not offer simple rules about how-best to play the game. Your friends are right to think about this as, at root, an ML problem.